Gap Probabilities and Betti Numbers of a Random Intersection of Quadrics

We consider the Betti numbers of an intersection of k random quadrics in RPn . Sampling the quadrics independently from the Kostlan ensemble, as n → ∞ we show that for each i ≥ 0 the expected ith Betti number satisfies: Ebi(X) = 1 + O(n−M ) for all M > 0. In other words, each fixed Betti number of X is asymptotically expected to be one; in fact as long as i = i(n) is sufficiently bounded away from n/2 the above rate of convergence is uniform (and in this range Betti numbers concentrate to their expected value). For the special case k = 2 we study the expectation of the sum of all Betti numbers of X. It was recently shown [27] that this expected sum equals n + o(n); here we sharpen this asymptotic, showing that: (1) n j=0 Ebj (X) = n + 2 √ π n1/2 + O(nc ) for any c > 0. (the term 2√ π n1/2 comes from contributions of middle Betti numbers). The proofs are based on a combination of techniques from random matrix theory and spectral sequences. In particular (1) is base...

• Type: Journal
• Year: 2016
• Where: DCG
• Authors: Antonio Lerario, Erik Lundberg
• Keywords: Computer Graphics, DCG 2016